Volume of Cones and Pyramids Geometry Unit 5, Lesson 8 Mrs. King

Reminder: What is a Pyramid? Definition: A shape formed by connecting triangles to a polygon. Examples:

Reminder: What is a Cone? Definition: A shape formed from a circle and a vertex point. Examples: s

Volume Of A Cone. Consider the cylinder and cone shown below: D The diameter (D) of the top of the cone and the cylinder are equal. H The height (H) of the cone and the cylinder are equal. If you filled the cone with water and emptied it into the cylinder, how many times would you have to fill the cone to completely fill the cylinder to the top ? 3 times. This shows that the cylinder has three times the volume of a cone with the same height and radius. www.ltscotland.org.uk/Images/volumesofsolids_tcm4-123355.ppt

Formulas Volume of a Cylinder: V = r2 h Volume of a Cone: V= 1/3 pr2h

Example #1 Calculate the volume of: V= 1/3 pr2h V= 1/3 (p)(7)2(9) V = 147pm3

Example #2 Calculate the volume of: V= 1/3 pr2h V= 1/3 (p)(5)2(12) V = 100pcm3

Example #3: r = = 2 V = πr 2h V = π(22)(7) An ice cream cone is 7 cm tall and 4 cm in diameter. About how much ice cream can fit entirely inside the cone? Find the volume to the nearest whole number. r = = 2 d 2 V = πr 2h 1 3 V = π(22)(7) 1 3 V ≈ 29.321531 About 29 cm3 of ice cream can fit entirely inside the cone.

Compare Compare a Prism to a Pyramid. Make a conjecture to what the formula might be for Volume of a Pyramid.

Formulas Volume of a Prism: Volume of a Pyramid: V = 1/3 Bh

Example #4 Calculate the volume of: V = 1/3 Bh V = 1/3 (102)(15) 15” V = 1/3 Bh V = 1/3 (102)(15) V = 500in3 10”

Example #5 Find the volume of a square pyramid with base edges 15 cm and height 22 cm. Because the base is a square, B = 15 • 15 = 225. V = Bh 1 3 = (225)(22) 1 3 = 1650

Example #6 Find the volume of a square pyramid with base edges 16 m and slant height 17 m. The altitude of a right square pyramid intersects the base at the center of the square. Because each side of the square base is 16 m, the leg of the right triangle along the base is 8 m, as shown below.

Example #6, continued Step 1: Find the height of the pyramid. 172 = 82 + h2 Use the Pythagorean Theorem. 289 = 64 + h2 225 = h2 h = 15 Step 2: Find the volume of the pyramid. V = Bh 1 3 = (16 x 16)15 = 1280